The problem asks us to find the value of $a$ in the quadratic equation $2x2 - 3x + a = 0$, given that its roots are in the ratio $1 : 2$.
Using Vieta's Formulas
Let the roots of the quadratic equation $Ax2 + Bx + C = 0$ be $p$ and $q$. According to Vieta's formulas:
- Sum of roots: $p + q = -B/A$
- Product of roots: $p * q = C/A$
In our equation, $2x2 - 3x + a = 0$, we have $A = 2$, $B = -3$, and $C = a$.
Applying the Ratio of Roots
We are given that the roots are in the ratio $1 : 2$. Let the roots be $k$ and $2k$.
Sum of roots:
$k + 2k = -(-3)/2$
$3k = 3/2$
Solving for $k$:
$k = (3/2) / 3$
$k = 1/2$
Product of roots:
$k * 2k = a/2$
$2k2 = a/2$
Calculating the Value of 'a'
Now, substitute the value of $k = 1/2$ into the product of roots equation:
$2 * (1/2)2 = a/2$
$2 * (1/4) = a/2$
$1/2 = a/2$
Multiply both sides by 2:
$a = 1$
Therefore, the value of $a$ is 1.
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